Note
\( \int \phi (\tau ) e^{it\tau } \big \langle \lambda ^* \chi _+^{-k}, \widehat{f}(\theta ,\tau ) e^{iy\cdot \theta } \big \rangle _\theta d\tau = 2\pi \int \phi ^{\vee } (t-s) \big \langle \lambda ^* \chi _+^{-k}, \mathcal {F}( f(\cdot ,s))(\theta ) e^{iy\cdot \theta } \big \rangle _\theta ds, \) 
where \(\mathcal {F}\)  denotes the \((d-1)\) -dimensional Fourier transform. Hence we observe that
\(\int \phi (\tau ) e^{it\tau } \big \langle \lambda ^* \chi _+^{-k}, \widehat{f}(\theta ,\tau ) e^{iy\cdot \theta } \big \rangle _\theta d\tau = (2\pi )^d \int \phi ^{\vee } (t-s) T_k(f(\cdot , s))(y) ds,\) 
where \(T_k\)  is the Bochner–Riesz operator of index \(-k\)  defined in (REF ) with \(n=d-1\) . We also use the following simple inequalities (see, e.g., [1]})
\(\Vert f\Vert _{L^{q,\infty }_x}\le \Vert f\Vert _{L^q_tL^{q,\infty }_y}\quad \text{and} \quad \Vert f\Vert _{L^p_tL^{p,1}_y}\le \Vert f\Vert _{L^{p,1}_x}.\) 
Making use of the equality (REF ), the first inequality in (REF ), and Minkowski's inequality,
\( \bigg \Vert  \int _{\mathbb {R}}\phi (\tau ) e^{it\tau }\big \langle \lambda ^* \chi _+^{-k}, \widehat{f}(\theta ,\tau ) e^{iy\cdot \theta }\big \rangle _\theta d\tau \bigg \Vert _{L^{q,\infty }_x} \lesssim \bigg \Vert  \int _{\mathbb {R}}\phi ^\vee (t-s) \Vert  T_k(f(\cdot , s))\Vert _{L^{q,\infty }_y}ds \bigg \Vert _{L^{q}_t}. \) 
Applying Young's inequality, the estimate (REF ) with \(n=d-1\) , and the second inequality in (REF ), we see that
\( \bigg \Vert  \int _{\mathbb {R}}\phi ^\vee (t-s) \Vert  T_k(f(\cdot , s))\Vert _{L^{q,\infty }_y}ds \bigg \Vert _{L^{q}_t}\lesssim \Vert  \phi ^\vee \Vert _{L^r} \Vert  f\Vert _{L^{p}_tL^{p,1}_y}\le \Vert  \phi ^\vee \Vert _{L^r} \Vert  f\Vert _{L^{p,1}_x} \) 
for \(r\in [1,\infty ]\)  satisfying \(1+\frac{1}{q}=\frac{1}{p} +\frac{1}{r}.\)  Since \(\mathrm {supp\,}\phi \subset [-2,2]\) , we have \(\Vert \phi ^\vee \Vert _r\lesssim \Vert \phi \Vert _{C^2}\)  for every \(r\in [1,\infty ]\) . Therefore, we obtain (REF ).
The \(L^{p,1}-L^q\)  estimate for \((\frac{1}{p}, \frac{1}{q})\in (B_{k}^d, D_{k}^d]\)  can be proved similarly by using the estimate () in place of (REF ) (with \(n=d-1\) ).
